Riemann–Hilbert approach to the focusing and defocusing nonlocal complex modified Korteweg–de Vries equation with step-like initial data

Abstract

Recently, research about nonlocal integrable systems has become a popular topic. Here, we mainly use the Riemann–Hilbert (RH) approach to discuss the nonlocal complex modified Korteweg–de Vries (cmKdV) equation with step-like initial value. That is the Cauchy problem, i.e., we establish the analytical relation between the solutions r(z, t), r(−z, −t) of the nonlocal cmKdV equation and the solution of a matrix RH problem. First, we analyze the eigenfunctions of the linear spectral problem of the nonlocal cmKdV equation. Second, we discuss the scattering matrix T(ɛ) and its spectral functions α1(ɛ), β(ɛ) and α2(ɛ) depending on the prescribed step-like initial value. Finally, we find that the solution of the Cauchy problem of the nonlocal cmKdV equation can be represented by the solution of the corresponding matrix RH problem.